• Title/Summary/Keyword: differential inequalities

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ON RETARDED INTEGRAL INEQUALITIES OF BIHARI-TYPE

  • Choi, Sung Kyu;Choi, Taeyoung;Kim, Daejung;Koo, Namjip
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.1
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    • pp.49-63
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    • 2009
  • We obtain some retarded integral inequalities of Bihari-type and apply these results to a retarded differential equation of Bernoulli-type.

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SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES

  • KIM, YOUNG JIN
    • The Pure and Applied Mathematics
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    • v.22 no.4
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    • pp.315-331
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    • 2015
  • The purpose of this paper is to obtain Opial-type inequalities that are useful to study various qualitative properties of certain differential equations involving impulses. After we obtain some Opial-type inequalities, we apply our results to certain differential equations involving impulses.

Two-Weighted Intergal Inequalities for Differential Forms

  • Xiuyin, Shang;Zhihua, Gu;Zengbo, Zhang
    • Kyungpook Mathematical Journal
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    • v.49 no.3
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    • pp.403-410
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    • 2009
  • In this paper, we make use of the weight to obtain some two-weight integral inequalities which are generalizations of the Poincar$\'{e}$ inequality. These inequalities are extensions of classical results and can be used to study the integrability of differential forms and to estimate the integrals of differential forms. Finally, we give some applications of this results to quasiregular mappings.

RETARDED NONLINEAR INTEGRAL INEQUALITIES OF GRONWALL-BELLMAN-PACHPATTE TYPE AND THEIR APPLICATIONS

  • Abdul Shakoor;Mahvish Samar;Samad Wali;Muzammil Saleem
    • Honam Mathematical Journal
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    • v.45 no.1
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    • pp.54-70
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    • 2023
  • In this article, we state and prove several new retarded nonlinear integral and integro-differential inequalities of Gronwall-Bellman-Pachpatte type. These inequalities generalize some former famous inequalities and can be used in examining the existence, uniqueness, boundedness, stability, asymptotic behaviour, quantitative and qualitative properties of solutions of nonlinear differential and integral equations. Applications are provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problem for nonlinear integro-differential equation and Volterra type retarded nonlinear equation. This research work will ensure to open the new opportunities for studying of nonlinear dynamic inequalities on time scale structure of varying nature.

Basic Results in the Theory of Hybrid Casual Nonlinear Differential Equations

  • DHAGE, BAPURAO CHANDRABHAN
    • Kyungpook Mathematical Journal
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    • v.55 no.4
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    • pp.1069-1088
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    • 2015
  • In this paper, some basic results concerning the existence, strict and nonstrict inequalities and existence of the maximal and minimal solutions are proved for a hybrid causal differential equation. Our results generalize some basic results of Leela and Laksh-mikantham [13] and Dhage and Lakshmikantham [10] respectively for the nonlinear first order classical and hybrid differential equations.

THE THEORY AND APPLICATIONS OF SECOND-ORDER DIFFERENTIAL SUBORDINATIONS

  • Lee, Jun Rak
    • Korean Journal of Mathematics
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    • v.7 no.1
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    • pp.85-101
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    • 1999
  • Let $p$ be analytic in the unit disc U and let $q$ be univalent in U. In addition, let ${\Omega}$ be a set in C and let ${\psi}:c^3{\times}U{\rightarrow}C$. The author determines conditions on ${\psi}$ so that $$\{{\psi}(p(z),zp^{\prime}(z),z^2p^{{\prime}{\prime}}(z);z){\mid}z{\in}U\}{\subset}{\Omega}{\Rightarrow}p(U){\subset}q(U)$$. Applications of this result to differential inequalities, differential subordinations and integral inequalities are presented.

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